Spin transport revealed by the spin quantum geometry

We present the framework of \textit{spin quantum geometry}, which is fundamentally linked to the spin degree of freedom of Bloch electrons and incorporates both the spin quantum geometric tensor (QGT) and the recently introduced Zeeman QGT, to elucidate the spin transport. We show that the spin and Zeeman QGTs, respectively, provide a unified framework for revealing known spin currents, such as the intrinsic spin Hall effect, and spin magnetization, such as the Edelstein effect, of Bloch electrons under an electric field. In addition, we predict the linear displacement spin Hall effect, wherein an AC electric field induces a transverse spin current in insulating systems. Furthermore, we propose two novel nonlinear spin responses: the nonlinear Drude spin current (NDSC) and the nonlinear Drude spin magnetization (NDSM), both of which exhibit a quadratic dependence on the relaxation time, like the nonlinear Drude charge current, and are governed by the \textit{spin quantum geometry}. Finally, we evaluate the NDSC and NDSM with Dirac models of topological insulators and find that, in the moderately dirty regime, the NDSC and NDSM can exceed their respective nonlinear intrinsic counterparts, which have recently garnered significant interest in spintronics.